Identifier
Values
[1] => [1,0] => [1] => 0
[1,1] => [1,0,1,0] => [2,1] => 1
[2] => [1,1,0,0] => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 2
[1,2] => [1,0,1,1,0,0] => [2,1,3] => 0
[2,1] => [1,1,0,0,1,0] => [1,3,2] => 0
[3] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 3
[1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 0
[4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,7,2] => 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,3,4,5,6,2,7] => 0
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,3,4,5,2,7,6] => 0
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,3,4,5,2,6,7] => 0
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,3,4,2,6,7,5] => 0
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,3,4,2,6,5,7] => 0
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [1,3,4,2,5,7,6] => 0
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => [1,3,4,2,5,6,7] => 0
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,5,6,7,4] => 0
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,5,6,4,7] => 0
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,7,6] => 0
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,5,4,6,7] => 0
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [1,3,2,4,6,7,5] => 0
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2,4,6,5,7] => 0
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,3,2,4,5,7,6] => 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [1,3,2,4,5,6,7] => 0
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [1,2,4,5,6,7,3] => 0
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,2,4,5,6,3,7] => 0
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [1,2,4,5,3,7,6] => 0
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [1,2,4,5,3,6,7] => 0
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [1,2,4,3,6,7,5] => 0
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5,7] => 0
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,2,4,3,5,7,6] => 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [1,2,4,3,5,6,7] => 0
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [1,2,3,5,6,7,4] => 0
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,2,3,5,6,4,7] => 0
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,2,3,5,4,7,6] => 0
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [1,2,3,5,4,6,7] => 0
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [1,2,3,4,6,7,5] => 0
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,2,3,4,6,5,7] => 0
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 0
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Description
Minimum over maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the minimum of this value over all cycles in the permutation.
For example, all permutations with a fixed-point has statistic value 0,
and all permutations of $[n]$ with only one cycle, has statistic value $n-1$.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
bounce path
Description
The bounce path determined by an integer composition.